Shapeways: Bring your creativity to life in 3Dhttp://www.shapeways.com/forum/index.php
Shapeways: Bring your creativity to life in 3D

Somsky Gears by BILL SOMSKY and OSKARhttp://www.shapeways.com/forum/index.php?t=rview&goto=59118&th=12128#msg_59118
Bill Somsky invented an algorithm to design exactly fitting Looney Gears, which I would call Somsky Gears. The key understanding is to start with three gears on a diagonal inside the fourth, e.g. 37=7+13+17. Bill proved mathematically that this line of gears can flop around inside the larger circle while retaining an exact fit. Then repeat the process for another set, keeping the outer and central gear the same, e.g. 37=11+13+13. The final step is to overlap the two sets such that the middle gear of one set coincides with the middle gear of another set. This can be achieved only at discrete positions, namely Somsky Gear positions.

The explanation may be a bit hard to follow, so I hope that the video and illustrations help. The Somsky algorithm can help finding exact sets with four planetary gears. Bill even found a set with six planetary gears, which he expects to be very rare.